Normalized defining polynomial
\( x^{10} - 2x^{9} + 5x^{8} - 6x^{7} - 2x^{6} + 10x^{5} + 14x^{4} + 15x^{3} + 10x^{2} + x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(351721503125\) \(\medspace = 5^{5}\cdot 103^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(14.28\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $5^{1/2}103^{1/2}\approx 22.693611435820433$ | ||
Ramified primes: | \(5\), \(103\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{417}a^{9}+\frac{17}{139}a^{8}+\frac{67}{417}a^{7}-\frac{23}{139}a^{6}+\frac{94}{417}a^{5}+\frac{127}{417}a^{4}-\frac{205}{417}a^{3}-\frac{49}{139}a^{2}+\frac{142}{417}a-\frac{118}{417}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{102}{139}a^{9}-\frac{796}{417}a^{8}+\frac{2015}{417}a^{7}-\frac{3044}{417}a^{6}+\frac{414}{139}a^{5}+\frac{2305}{417}a^{4}+\frac{2878}{417}a^{3}+\frac{2834}{417}a^{2}+\frac{445}{139}a+\frac{57}{139}$, $\frac{113}{417}a^{9}-\frac{214}{417}a^{8}+\frac{482}{417}a^{7}-\frac{430}{417}a^{6}-\frac{776}{417}a^{5}+\frac{660}{139}a^{4}+\frac{1021}{417}a^{3}+\frac{1876}{417}a^{2}+\frac{252}{139}a+\frac{10}{417}$, $\frac{184}{417}a^{9}-\frac{485}{417}a^{8}+\frac{1208}{417}a^{7}-\frac{618}{139}a^{6}+\frac{755}{417}a^{5}+\frac{1406}{417}a^{4}+\frac{1756}{417}a^{3}+\frac{575}{139}a^{2}+\frac{45}{139}a-\frac{102}{139}$, $\frac{125}{417}a^{9}-\frac{436}{417}a^{8}+\frac{336}{139}a^{7}-\frac{1675}{417}a^{6}+\frac{908}{417}a^{5}+\frac{1558}{417}a^{4}+\frac{30}{139}a^{3}-\frac{1139}{417}a^{2}-\frac{598}{417}a-\frac{98}{139}$, $\frac{422}{417}a^{9}-\frac{718}{417}a^{8}+\frac{1864}{417}a^{7}-\frac{1874}{417}a^{6}-\frac{1615}{417}a^{5}+\frac{1370}{139}a^{4}+\frac{7037}{417}a^{3}+\frac{7883}{417}a^{2}+\frac{6131}{417}a+\frac{1912}{417}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 78.0020419672 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 78.0020419672 \cdot 1}{2\cdot\sqrt{351721503125}}\cr\approx \mathstrut & 0.409973684318 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.10609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.1312418339203244062978515625.1 |
Degree 10 sibling: | 10.0.36227314821875.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(103\) | 103.2.0.1 | $x^{2} + 102 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.103.2t1.a.a | $1$ | $ 103 $ | \(\Q(\sqrt{-103}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.515.2t1.a.a | $1$ | $ 5 \cdot 103 $ | \(\Q(\sqrt{-515}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.2575.10t3.a.b | $2$ | $ 5^{2} \cdot 103 $ | 10.2.351721503125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.103.5t2.a.a | $2$ | $ 103 $ | 5.1.10609.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.103.5t2.a.b | $2$ | $ 103 $ | 5.1.10609.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.2575.10t3.a.a | $2$ | $ 5^{2} \cdot 103 $ | 10.2.351721503125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |